Combine the water, butter, and salt in a medium-sized saucepan, heat until the butter has melted, and bring to a rolling boil., Remove the pan from the heat, and add the flour all at once, stirring vigorously., Return the pan to the burner and cook over medium heat, stirring all the while, until the mixture smooths out and follows the spoon around the pan; this should take considerably less than a minute., Remove the pan from the heat, and let the mixture cool for 5 to 10 minutes. It'll still feel hot, but you should be able to hold a finger in it for a few seconds. If you have an instant-read thermometer, the temperature should be below 125°F., Transfer the mixture to a mixer, and beat in the eggs one at a time. The batter will look curdled at first, but when you add the last egg it should become smooth. Beat for at least 2 minutes after adding the last egg., Pour a generous 4 cups vegetable oil (peanut oil preferred) into a 10" electric frying pan or heavy skillet set over a burner; the oil should be about 5/8" deep. If you use a smaller or larger pan, add oil to a depth of between 1/2" and 3/4". Heat the oil to 375°F., Scoop small (1") balls of batter into the hot oil, filling the pan but not crowding the doughnuts. A teaspoon cookie scoop, filled level, works well here., Fry the doughnuts for about 6 minutes. As they cook they'll turn themselves over, usually multiple times. Use a chopstick or pair of tongs to give a nudge to any that seem to be stuck on one side. After about 6 minutes, the doughnuts should be a deep golden brown., Transfer them from the frying pan to a paper towel-lined baking sheet to drain and cool. Repeat with the remainder of the batter., Use a pastry bag equipped with a plain tip to pipe jelly into the interior of the cooled doughnuts. Shake them gently in a bag with granulated or confectioners' sugar. Serve immediately; or wrap loosely and store at room temperature. The doughnuts will gradually soften as they sit.